Discussions, incl. FW Lawvere, M Wright

0:00 Just that one personal property. Performing it in some way, I always find this quite unbelievable. It's enough to actually prove that this right adjoint actually is the desired left adjoint.

2:30 That's kind of hard to see, but actually it is. Psychological ones have been repeatedly, repeatedly told. This never actually, any attempt at formalization of this alleged construction. All of these terms can be used to define a set of functions that can be used to define a set of functions that can be used to define ...somehow exists and is the real guarantee of when you change categories or change the universe because it's not real in a way. It's illusory. So for that reason I think every philosophy based on this naive idea of constructivism is ultimately false, be it Kronecker or Brouwer or who else.

5:00 Brouwer's a strange one, you see, because he seems like he knew this, and yet... So he was a more subtle subjective idealist. By claiming to be a philosopher, he seemed to be a more extreme, more true, naive subjective idealist. But as a philosopher, as an mathematician, he seems to be... He's apparently actually stated that he thinks he himself is the only living soul in the world, at one point in his career, anyway. That's about as strange as you can get. But on the other hand, the mathematical form of it was called to deny the natural number object. But it's very subtle because somehow in practice, he really reinforced the natural number object. I cannot quite spell out how that happened. It's like Hitler and Mussolini arguing against the capitalism only in order to reestablish it. By the way, he was a supporter of the Nazis. Wow! I'm afraid you're wrong. So that's what he's doing, exactly. He's doing lots of propaganda against this. ...naive completion of subjective infinity, because he knows you'll get popular support. And in the final analysis, his actual conclusion is even more subjective idealist. I never thought of this, a direct, not only analogy, but even coupling with the... Well, there have been a lot of, you know, recruits. It's made his expense, because, you know, he met his end because of an auto accident. No, he was actually knocked down and killed. So he was a great agent by 1986. He was knocked down and killed by some of his students, and that must have been pretty tough for a solipsist. It's also quite interesting that part of the talk that was given to me this week was about Brown's relations with Heidegg,

7:30 and also Brown's work as a philosopher, well actually it was. There was one speaker who claimed that there was evidence that another, a German man, had actually done this some three years earlier, but had not accomplished any of it. A certain man has written that some evidence did indeed have been in the university library in Amsterdam, and that Raymond Brown had a job there arranging the box set. This is a finger of suspicion at this point, but I have no idea what it is. I've got to tell you, I don't quite understand the whole point. Kimbrough's crazed, sort of, solitary philosophy. He couldn't really get Houston to play for him. He didn't create, necessarily, the mens rea. How could he plagiarize? He was the only mind in the world. So, serious point. Now, the serious point is the one you make, that this version of very strident, extreme subjective ideas in philosophy, coupled by a much more subtle, and, in mathematics, the same, well, position with respect to natural numbers, and this needs to be kind of flattened a little bit. The objective idea is... Trotskyite friend, Moshimako, he once, I remember he lectured once, you know, it has some certain possibility that you have an approximate x-part because, you know, things are fuzzy, you know, it's sort of under the guise of constructivism, but you could also interpret it in other ways.

10:00 You don't really need completeness of the metric space if you're only looking for an approximate point. The existence of the point as a precise point is similar, not at all identical to, but it's similar to the actual existence of open field or something of that sort. The approximating entity is much more solidly established than the point itself, and in practice that's all you need. So if we go on to American pragmatism, it's a very Trotskyite position. Well, I mean, I'm not even saying this because it's not Trotskyite, but yes, it was somehow. Well, you can work with Robin, he's been retired now for some years, and he's an academic for a while now, and he works in the mathematical voting system. It seems to be getting more and more popular. Yes, it publishes in economics journals for some of the economists. No, no, it's in fact been changed to John's. It's published in John's book, if that makes sense. The interesting point is the position in the founder of...

12:30 Moody's program is very important, yes, and the Israeli-Palestinian are highly decorated as a soldier in the Israeli army in 1956 and served again in 1967, quite seemingly. Then shortly after 1967, they were completely, and so much so, legal mechanism for the series, I've actually been sitting in a restaurant several years ago where there were some young students who had no idea who they were. I don't make quite a list of everything I've told you, but this is what there was in the West, so it was impossible not to hear what they were saying. And they were talking about other things, and they obviously knew personally, or their parents were one of them, clearly knew very well. One of them said, and this was at the time of some of the worst things in science, intelligence. Whenever I find myself starting to make decisions, I stop and think of what should matter, and that's what stops me from becoming Jerry. I thought that was a pretty good thing to have said about human being.

15:00 But how striking would it have been for us to be able to see how it was that the geometers should have responded at the time when these pathological functions were presented to them and they were told, well of course this means that there can't be a foundation among them. I was even given a stoic nature issue, because the one through the atomic heaven everything is completely different. With Powell on this point, one is very worried that abolitionism is going to run a paradigm shift, roughly before 1900. But previously it meant geometrical evolution, and then it became objective evolution.

17:30 It's actually, I don't know, Madison Avenue, Flash, Goebbels, you know, in the sense that, in the sense that it does not simply argue with some position, it starts with the opposite position in order to obtain as much mass support as possible, and then counts on the relative ignorance of the masses to transform that effortlessly into... He'll deal with calculus, or I'm saying, and Mussolini before him, but even more, right, specific, right, and Mussolini even tried to turn the trick twice, even at the very end, after he had been overthrown, but he tried to turn it in, you know, he said, that's his clique, and it's going to come out, but I don't know, I don't know, the royalists, I don't know, I'm trying to save them. He then noticed himself every year as a group of fascist dictators, building up, acting as the instrument of, you know, raising the rate of exploitation of this cult. He then turned around at the very end and tried to turn things again as a kind of ultra-leftist, because of the rhetoric of the so-called Italian Social Republic. Which is the puppet regime, which is only described as a puppet regime, it was actually something a little bit more, I think, a lot more interesting than that, which was set up after the United States of America's springing of the Republic of Italy, the monstrosity of the Gideon. But his rhetoric was an extreme ultra-left, and his ultra-left is right in essence.

20:00 And, you know, there are a few historical illustrations of that more, without explicit, than the Republic of Sarlat, who had these streams of anti-Mortimer, anti-Mortimer, anti-Peregrine rhetoric, serving as the, as the auxiliaries to the SS? No. No, I think it was, I think it was, it wasn't the minor who said that, actually. That's an example, I mean, that's a partial expression of what I just said. Absolutely. In other words, in fact, you can even recognize who are actually the significant ruling class philosophers, by the fact that they really do use, they understand and use this technique, where a secondary philosopher is a naïve plagiarist. Yes, yes. Yeah, that's a criterion for recognizing which philosophies are not mentioned in the book. So, for example, Mirandula led his opposition to zoology. Or, Tommaso di Campanella, your initial introduction to him is, oh, he actually supported Galileo when all the others... That's how I've been introduced to him. Even though Galileo himself didn't really want this kind of support, it's sort of like, is Peter Johnstone going to refuse the Pachyderm Award? No, no, I mean it wouldn't be, I mean it's obviously incredibly inappropriate, but publicly to refuse it would be different from quietly accepting it. And that seems to have been the relation with Galileo too. For his own reasons, as a successor to Arnold Greenwald, he was built in many places.

22:30 The other man at that period, I would like to break in on, in the background, if you look at the background, it's kind of there. I think that they had a different take on Celio de Trento than any other. Interesting. He's one of the heroes, you know, of Capo di Fiori, where Giordano Bruno was hanged, and there's a statue put up after that. ...put up after 1871 by the Italian nationals, but you see there's a statue, but then around the statue there are, what do you call them, pictures, around the medallions, you know, various, lesser, but... These are important typicals of the 19th century, and that's the first time I ran across Paulus Hawking, and I was told right away that he was the woman who wrote a report on the counting of threads, different from all the others. Oh, and Lyndon LaRouche, Paul's hymn. You know, the basic, you know, traitor of all times. Oh! Well, in that case, there must be something to be said for Thomas Sophe there.

25:00 Linden LaRouche is really crazy. Oh, completely crazy. No, but it's strange. Sometimes he has, you know, he's got staff doing research all the time, right? So sometimes he comes up with some incredible stuff that you never heard of before, and part of it's actually true. My point is figuring out which part. Well, Sarky-Waterland was not a plagiarist. I've always wanted to understand more about him. He was from Venice. Indeed, when he was the great champion of the Venetian state, he was the one man, the think tank of the Venetian state. That was why he set off the trench, to make a sign of Washington on behalf of the Venetians. It's really good, isn't it? That's all right now. That's all right. I'll get me a coffee and... I'll go down and check out. Oui, oui, oui. Deux cafés. Deux cafés. Deux cafés. Let's just do more lighting. Why are you getting the relaxation? But it's very considerable claims are made by scientists and mathematicians. Oh, I didn't know that. Yes, and I would like to know more about them. But the claims that I've seen made were in 19th century books that were written by people who were not themselves, clearly not themselves, you know, qualified mathematicians, but it's said that he had a correspondence with Stevin when he was a theatre, or something like that, or something like that, Stevin, when he met him, allegedly.

27:30 He's interesting in that he seems to have represented a resistance by the most progressive I don't know whether he ever fast-forwarded with Galileo. I think he started with Galileo about 6,000 years ago. But he, of course, just like Galileo, forward-powered the whole of Venice, Because of their rejection of the contemplative claims of the peasant, and also because they allow the degree of religious toleration of the merchant makers, it's great in their interest to have a Protestant leader, you know, a merchant in their village, at Jurek.

30:00 I wish I was going to say, which would that it came again. I think there are indications that it is coming again. No, that's the problem. Witness the struturalism, so-called struturalism in the model. Structuralists who have no theory of structure, or who neglect the one really serious theory of structure. So the other thing, what Colin pointed out, the meaning of the word itself, between geometric intuition and the fence of geometry, and something often I can come up with that can't be the fence of the extreme, that is the interior of knowledge, Volcano.

32:30 Yeah, so I was corresponding. There is no trace of any influence or causality on training, and it's tied to the similarities of their minds. I doubt it very much, too. One interprets influence in terms of this narrow bourgeois definition of their own existing individuals. Good, and of course, maybe you could doubt it. You could anti-document it, i.e., you could show that they don't exist in the document. Well, of course, it's crazy. This is like, you know, this is like the mad, extreme neo-Nazi historian who was, um... And if you think in terms of the, uh... Because Hitler couldn't have been responsible for the Holocaust because there's no documentation proving it either. Yeah. No, but I mean, uh, in terms of a more, more adequate philosophy of the development of thinking, specifically one which recognizes, at least, the dialectical, collective, and individual. There's no way of proving that there's no, there's no end to it. Well, I continue to think of them as part of the same line, even though it's kind of just one of those barren of all these lectures, you know, spending your time trying to see, to show who read who and what they were talking about. Apparently Dedekind did read both of them. Between the first and second edition of his book. He's a friend of mine, Walter Felsher, now deceased. I made a special trip to the archive in Brownschweig to check that point, to see if the second piece was fake, to see if it was true, and then I could change the reading of these two editions because of the reading of Bozzano. What do you mean, what do you mean, what do you mean, what do you mean, what do you mean?

35:00 It's a strange, strange way that Jachin was so strong to set this combination of the continuum, which all cohesion has been removed, and just as it would have been ground down to dust particles. Yeah, I mean, this, what I'm saying is perhaps another instance of... You know, in the absence of explicit dialectical materialist defense, even the greatest are worn away by the attacks of idealism and are ultimately led to acceptance. They don't quite know why. I forget the exact point which we changed in this book. I would like to, I really would like to understand that. It's one of the well-known points in the book, and I would like to understand it. But I remember when you and I were in Florence, in Alberta, I think it was in 2000, I remember, it was just after. Who's done the work? That's right. It was also just after you'd spoken in Paris that we finished the lecture in Milan, one hour, along the other. We really had a very interesting conversation along the way. David came to ask Bob to set this version to continue from the talk that he used in London. I suppose because he didn't have to necessarily. Concepts are too rigorous. They're making a specific kind of connection. I'll take you back to the 21st century class. It's a good culture. It came across an awful culture on the internet.

37:30 He died two years ago. There remains these now. I haven't come across them. And they're still accessible, are they? No, they remain. They may remain for years without being drained. Good, good, good. There's some debate that a few years before he dies, he pops up and there he is talking. I mean, he was a scholar, a historian of mathematics, or... No, he, well, I don't know. He was one of the... Oh, God, I think I met him in the early 60s in Berkeley. He was a German from, I think he was from Frankfurt. Oh, he was from Frankfurt. And he taught in Freiburg and Breisgau. And then after that in Tübingen, he became, essentially, dean of Tübingen. There's at least one, you know, one or two interesting students, Schumacher, a fellow who told me about this point, he was part of my original institute at Dalhousie in Halifax, along with Miles Tierney and 12 others, yeah, he was there in that time, he has a book on two-volume work, Numbers and Logic, something of that sort, in German, but actually, You know, it's historical citation. So this red was a little interchange on the internet that I found was referring in particular to a historical point of view. Precisely this one. Yes, this is independent. Yes, he was actually saying, yes, he was able to document, I must find, I hope I haven't lost my copy of this book, he was able to document precisely this fact that Dedekind resisted.

40:00 I think it was either the Fomers, or I think it was maybe even the Fomers, or some other... There is also a discussion of combinations of new and old, read to the end of the regime, and what is out. If the Vatican did not consider it as a construction, well, you might as well call it that. Is there a F-E-L-S-E-A-G-R? F-E-L-S-E-A-G-R. I'm a student who wrote a book in German on the crisis of geological history. There is such a thought, and I was trying to get my hands on it a couple of years ago, but in any case I was able to trace it in an abacus. They had one copy and I sent it off to Britain, or even sell it to somebody else, but it seems I have a printer, and of course if you read, I don't have the terms to weigh through it by my own, but it is not worth it, but I think it would be incredibly worth it. I think it was published by Sprinter, I'm surprised that they didn't allow you to go out and print. Go inside? No, no, I've got to go.

42:30 Well, shall we get the bill and I'll stroll down to the car. It should all work together, and in topology, a large part of topology, which we're now continuing from topology to topology, and not just the fixed ones, not just the fixed ones in topology, nowadays that is regarded as a mere corollary of the great topology. Now, the great topology was developed largely by that use of the tree. But when I say hidden from history, I mean, for example, even MacLean didn't know about it. MacLean had a completely false theory of how homology turned out to be brute, because, you know, drastically terrifying stuff. It wasn't just these, I mean, Merkel, it was the third person. What's her name? You mentioned the Oscar guy. The Oscar guy. I've read that paper. Vietoris. Of course, Vietoris. Vietoris. The only reason I couldn't think of is you said Austro, so Vietoris doesn't sound like an Austrian name on there. No, probably. It sounds more like an Italian name. Yeah. I guess Vietoris. Yeah. It sounds like a three-or-one name. Not Italian, either. It's very like... Vietoris. Yeah. The Vietor... Well, there's really just three of them. The Vietoris and Mercer. And Pop. And it's published. In the royal decrees, you had to dig very far to find at least an outline of what you had done, but you had to know to look first. Nobody realized that there was such a model, not that I knew of.

45:00 McLean didn't know it, but it wasn't well known. Who was it who first introduced the notion, I mean, under that name, the notion of the vibration of the universe? Half-vibration? Not specifically, half, but the general notion of the universe. Well, the half-vibration is specific. Yes, yes, exactly. Namely, it's because the vibration of the universe, because it's three-stripped, it's two-stripped, it's because... Because the quaternions, the unit quaternions, of course, act on the three spheres, sorry, they are the spheres, the unit quaternions form the sphere. If you choose any particular point on the sphere and follow the action of the orbits ray, you get another sphere below. So there's this, there's this map between the two spheres that arises precisely because of the geometric means. It turns out to be that vibration also is one of the terms that in particular plays an enormous role. And then there's a generalization of the 17 spheres powered over by some other sphere, because an analogous sort of thinks that sometimes this whole sequence is called topofibration. It starts with the three-sphere, and the two-sphere, and the three-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere, and the two-sphere.

47:30 The answer is that he claimed to solve the problem of Poincare. It's in a way the reason why the problem of classifying all three manifolds is really solved in a satisfactory way. It became an obsession with apologists. It goes out of the mechanics. Yerkov worked actually in mechanics and he was talking about the problem of Poincare and mechanics. Systems of two degrees of freedom. Let's say Hamiltonian systems of two degrees of freedom. Therefore the phase space is four dimensional. But actually the energy is constant and so there's sort of an ellipsoid or a sphere of something like this of dimension three because the values of the Hamiltonian are one dimensional so there's The three-dimensional manifold living above the two-dimensional space, so that projection map is the Ur-Weisspiel of the vibration. Well, it's not particular because we took any two system of two degrees, any Hamiltonian constant of two degrees of freedom. I suppose there's some sort of a positivity constraint, whether that be constant or manifold or compact or something like this. But then, you see, so the desire to classify all possible dynamical systems of freedom, in effect... A large part of it would be to classify all three-dimensional manifolds, because any three-dimensional manifold might arise at the level set that the Hamiltonian function would meet. So this is very concrete and having to do with motion, laws of motion, not just pure being, but laws of becoming. So that example, just before 1920, became the inspiration. I think the first person to give a general definition of vibration.

50:00 There are vibrations by means of spheres, which in fact, again, the example would be like that, typically, because if the energy surface is an ellipsoid, then the fiber would be a circumference, one of the circumferences of the ellipsoid would actually be a circle, so the fiber would be a sphere after the ellipsoid. The fibrations, these fibers are spheres, so if you like, smoothly parametrized families of spheres, parametrized by their base space, which might be a two-dimensional, fixed two-dimensional sphere. Of course, the fibrations property, namely that given a starting point, given a path in the base, there's a unique lift, at least it's just determinism. It's expressed by vibration copy, roughly speaking. Now, Whitney got into this because his teacher, Morse, Marston Morse, was actually a physician. It might have been even a student, as they're called, but in any case, Marston Morse was the last of the great topologists who actually knew some physics, who knew a significant amount of physics. And there's this poignant thing that his student, Whitney, was given to read over the summer. Notes of basic knowledge that Morse had given in his mid-twenties, and you know, for some reason or other he enjoyed the summer rather than, you know, he didn't actually learn any physics, I mean, you know, he always learned some physics, but in terms of having an adequate knowledge of the current state of classical physics in order to really apply these things, Morse was the one who did, and the student should have, but didn't. I remember once someone who and I had instructed on these facts was lecturing on MacLean's presence. He said, well, the problem was that Whitney didn't know any physics. Whereupon MacLean jumped with speed and says, how dare you say something like, a great man like Whitney, you don't know anything.

52:30 So I had to rise to the young man's defense and say, well, actually, I told him that, and the reason I told him that is because Whitney himself wrote it. I don't know any physics because of this summer experience, and so at that point, supposedly because of this subjective idealist reason that the young man was chasing girls, et cetera, or something, we don't know exactly what, but something other than studying physics, therefore the bifurcations between the progress of classical physics and the... Geometry of topology, yes. Which, of course, as you then said, went on on its own for a long time, doing good things. So then, you see, then there came sort of the Hough example, which is also a concrete example. It's a concrete algebraic example from the Quaternion. Yeah, I can see that. Specific construction, in fact, has marvelous properties of that nature. And moreover, because it could be continued through the integers, it gives it some concrete information about homotopy groups, which is hard to come by. I think this is sometimes the first non-trivial set of information that you get about all homotopy groups, because of the pulse vibrations, the analogous expression of that. But the general concept, you see, the general concept that there are, you see, there's Horowitz vibrations, Serre vibrations, Kahn vibrations. So, this was the early 50s, you know, they were all, they were all struggling. Of course, Kahn's definition is in the combinatorial context, this is what you're set, you know, like this. And so, you know, it's the sort of thing that you have a precise analog for cubical sets and other... Combinatorial toposes, I mean, sometimes there's not really a variable, it's just, you know, the thing that it should be in the analogous case and all these things are combinatorial toposes. On the other hand, Horowitz and Serre, well, this is up to homotopy, you won't be able to tell the difference, but the actual meaning in terms of actual geometry and analysis is another matter.

55:00 And they do differ in some ways. Courses on topology, in fact, Myles Tierney's course last week, that non-fibrations turn out to be the same thing, quote-unquote, as serifibrations, meaning that if you apply the geometric realization comfort of these combinatorial things, you will get serifibrations like that result. And I'd rather suspect that it would be equally true that you'd get the Raywich vibration. I'm not sure, but... The way which was, of course, more himself, actually more, he knew more about physics than these other people. So he was, again, closer to the old Birka and Morse. I go on and on, so I actually do something about history, don't I? Over the years, you get these, you put together these fragments. What about the neurophysicology? Right, right. This is something that I'm not familiar with. It's a specific treatment that I'm having. It's a combination of precisely my notion of out-tableing. Well, I was going to say, it's because... With homotopy. I've come across it in the context, precisely. With homotopy. So it was a standard way of resolving a space into, you know, analyzing it in terms of successive combinations, which were... This higher connectivity was more and more like a given space, but it was simpler and had a higher kind of coverage. Glad I know that. It's called Moore-Folstikoff because, I believe in this case, Folstikoff published first. First he was what he was, and then Moore had published something similar to this. It was then shown or construed that Moore's discovery was the same. More or less the same idea, and so after that, at least among Americans, it was known as Morai-Postakoff.

57:30 I tried to read Postakoff's paper, and it was translated into English, and it's extremely philosophical. It's very dialectical, so much so that I can't understand it. It's very compressed, but he clearly had a... There's a powerful vision of what this way of analyzing space is supposed to mean and what it didn't mean. So it comes up technically in spectral sequences and that sort of thing, because you're in one way of refreshing a space by analyzing it in a very simple way. In terms of a simpler space system, you calculate the homotopy or the homology or something, and then you pass through the limiter and that's what you get. I suppose it's viewed as a technical tool of computation, kind of. Vostokoff himself had quite another view. But exactly what that view was, I can't tell you. I've always meant to... This is one of the questions I was going to ask. I don't know. He was a genuinely Soviet mathematician of the Stalin era, or the immediately post-Stalin era, I'm not sure exactly what it is, but clearly an educator of the Stalin era. The first publication might have been in the KBV area, teaching mathematics. Nor do I know what ever happened to them, whether they were published or not. I suppose one can find out by asking Google. It seems to be quite an interesting figure, or its most probable approach to the mathematics. You know, which might ultimately be described as Platonist or something, I don't know, but at least it was conceptually more unified, apparently, from his own description, but not at all superficially. When did you hear about Kostnikov? From you. Oh, sorry. We were from reading Catherine's The Space and Quantum. Oh, yes. I mentioned that. You discussed it specifically in the context of homotopy, perhaps as illustrating the distinction

1:00:00 between Catherine's and... That could be candidates for spaces of quality, to be honest. So certainly the South Able was connected with the United States. It's very definitely part of it. Not just UIO, but the fact of climbing up, letting the left adjoint be absorbed by the next right adjoint, and nevertheless having both adjoints. Yeah, it's very much that, literally that, the co-skeleton. Probably he was the first one to explicitly use the co-skeleton because, you know, if you're just taking any sort of naive combinatorial point of view, you can imagine the skeleton. You see you have this thing and you throw away the higher dimensional pieces and you have left the skeleton. You have a solid tetrahedron, you throw away the middle, you've got a two-dimensional boundary, you throw away the spaces, you've got this one-dimensional framework, and so on, right? So this kind of idea of taking the skeleta as, you know, intuitively contained in the finite common theory, the corresponding co-skeletal discovery is not so intuitive. In fact, there's still many, many books on that. There are some things that we don't know precisely, but we do know precisely it's a three-skeleton. You can say, well then, what is it? Well, all we can say is, well, it must be a three-skeleton. What else can we do? No less. These things are rigidly determined. It must remain, you know, it may go on into time, but it stays a three-color phenomenon. On the other hand, the first color is just the opposite, because what are all the possibilities that it could become? So in some sense, these two are actually identical opposites.

1:02:30 Abstractly, the information is the same, but the way that it's plugged into the world of possible conceptual objects is totally opposite. One includes all the all the possibilities that are compatible with this partial information, whereas the other is, uh, is, uh... Well, there's an echo there, isn't there, of the relative distinction between the discrete and the co-discrete spaces? Oh, yeah, there is one. Because one is the one, and the co-discrete one is the other. Yes, yes, yes. One point can become any other in any possible way without any attention to the actual parametrization of the motion. You're right. In fact, it's literally in zero-dimensional case. Yes, of course it is. It is at the lowest level, of course, you're right. It's just that n equals zero. Of course, it's actually a special case. So you're right on the ball there. Now let's go to n equals one. n equals one is especially easier to visualize than any other one because it's still related to pi zero, to the components' function. But the components' function goes to the zero level and there is no corresponding components' function to the one or the two or the three levels. The out-pavement of zero, i.e. the definition of one dimension, is just the lowest of all possible levels, such that if you take the skeleton of that level as an arbitrary thing, then it still has exactly the same components as the arbitrary thing. So the one skeleton. Again, incredibly intuitive, if it does. The ones called from the consistent paths. That's what we need to ring up your craft. So I can define the one co-skeleton just in terms of sort of a dual of that concept, whereas the two co-skeletons already exist, but to give a more precise calculation is less clear.

1:05:00 I suppose you have the framework for a pyramid. The two dimensional, the two co-skeletons of it would... Well, does it just fill in the edges here, or does it fill in all the edges millions of times? What are really all the possible two-dimensional spaces between one skeleton that is given to us? They're all put together in one thing, all the possibilities that one has. It's the idea that all possible worlds exist simultaneously. All those that are possible with a given constraint exist simultaneously in the alphabets, in the co-spellings. But here, of course, one is dealing with the concepts and not reality. And also, of course, as one moves upwards. There are a number of different versions of this mathematical construction that can be involved more and more in making explicit the way, you know, the constraints of the way in which one at one point can become another, and that's clearly not the intention of the mathematical equation of the equation of zero, so it's not a construction. It's going to depend more on the actual category of deletion that we're in, whereas this lowest level description is sound. It sounds like it's completely absent here today. Well, it has to be the after piece, you could say, or this could be exactly the piece that's coming on. Yeah. No, I mean, even the what? The one, the next one, the description of the... Ah, bonjour! Pleasure to meet you. Even the description of the one, in terms of the components, you see, this is a more concrete description of the one you mentioned.

1:07:30 It means many different things. Concretely, it does depend on the category of the lecture, but at least the description is more vivid, whereas the next stage is going to depend largely on that. Sorry, the next stage? What do you mean? It will depend, you know, two, two. The two? Yes, it's going up. I mean, what is the surface? Yeah, that's almost a good, yes, that's good. Or rather, again, the co-skeleton means, given the fact that the arbitrary surface is the skeleton of some higher-dimensional space, what are the possibilities for that higher-dimensional space? So, they're broader because it's higher-dimensional. On the other hand, they're more narrow because you've given them quite a bit of constraint already by saying that you know the two-skeleton. The two-skeleton is the key component. Yeah. Yeah. Let's go. Yeah, and in my, in my . The universal property of the mind. Let's go, let's go. Let's go, let's go. Let's go, let's go. Let's go, let's go. Let's go, let's go. In these cases, the values of the line are entirely determined by the two-dimensional pieces of space.

1:10:00 Now, particular categories of cohesion are even sharper than the function of the map of the past. So you get this more precisely, but it's surprising to see that for any category, that the two-dimensional pieces of anything in the algebraic are enough to determine the one-dimensional functions. Normally when we talk about functions, these are always one-dimensional, the intensive qualities of dimension one. It's usually considered to determine everything in the algebraic. It's surprising. Whatever kind, whether it be analyticity, or smoothness, or continuity, or whatever it might be, the two-dimensional case of it, in every possible way, suffices for those kinds of values. I mean, universal property, quantifying these are all one-dimensional spaces, actually. You see, from my point of view, it's quite a... it sounds like a strong theorem. It only depends, it only becomes a mathematical theorem when you detect it, given the determination of what cohesive space means, given the calculation of what a two-dimensional space is, and so forth, which itself is part of the alphabulum. In a given case, in a given example, it could be that the alphabulum of one is infinity. You see, that would depend on a particular case. If that were true, then it's no surprise that it's a cultology. But I'm sure that it's always true. It comes with some crucial sense, for example. What I was saying, it's amazing how all these people take some crucial sense as a god.

1:12:30 And they don't know anything about particular properties. They just know how to calculate it. They all have their own intricacies and they think they're different from one another, and the horns and the mountain spears, and some kind of sort of sphere, and categorical properties, and they don't know any of it, even if they're categorical. Now the fact that there's some kind of spindle going on is already indicated in the notation, because they call simplistic sets S. They have dropped the particularity of it, whereas Godendieck wrote this whole 800-page manuscript trying to discover the question of should it be two-digit, or should it be globular, or should it be triangular, and all these kind of categories are more or less equivalent to a whole book of literature. So each of them has S to the delta of, or S to the box of, or S to the G of, or whatever. I mean, they all have some decoration. It indicates which kind of a combinatorial structure one is taking, but their anchors to conceal this fact is that they just call it S. It's very strange. It's even royal. In the old days, it was incredibly sharp at extracting the essence of a thing to make an inclusive action, and somehow succumbed to this, and then we go, once we've accepted that, we go into incredibly complicated things with it. But the fact that its own central role is to come to the fundamental question, we'd rather not deal with it.

1:15:00 Anyway, I'm just saying that in all these cases there's a sort of intuitive idea of what dimension N means. And dimension 2 does mean what it should mean. The definition based on alphabets doesn't give you the definition 2 that it's supposed to give you. Sorry, go ahead. That's the reason I mentioned it. But the remark that you made in your Cambridge lecture, if you look at the code of speech, in the case of the grammar of this course, is that there is no attention to the grammar of mathematics, and that you get by one point that you're coming up with precisely the opposite of those. Trivial. Great. And that shows up in the case of your discrete co-discipline spaces, closing down the complex and the part of the speculative space, when you treat this point as true. I understand. Closing down... Yeah. Compact. No, not compact. I was thinking, no, just the relationship between the components factor and the points factor in the, you know, the components. No, we haven't, well, no, no, sorry. It depends on if the universe itself can't work on space. Yeah, that's what I mean in that case. Well then, because both things are the identity. Yeah, yeah, so it seems... So in particular they're equal. Yeah, so it's trivial in this case, yeah. True, but that's more trivial than you actually mean, I think. No, the point is if you have a more or less arbitrary concept of cohesion, then there should be components that aren't all being the same. The antithesis is quality instead of cohesion. But the nature of the code of speed, the way the code of speed is opposite to, is identical with, but opposite to, discrete, is how you expect, because in that discussion the more general spaces don't directly come into it, even though they're in the picture.

1:17:30 So you start doing the alphabets. Again, quantified over all these stages, it must not be determined by the contradiction between the two sides of the next stage, but, as I was saying, the next stage, i.e. the one-dimensional state, is describable in the general way of algebra, but it's also describable in terms of the components that still exist at that level, and not at zero. There's nothing like it coming from infinity down to one. So you have only the unifying identity of opposites on this side that can overcome the other. So it would be great if we could come up with some way to calculate the outlabeling of dimension one, the so-called dimension two, outlabeling of dimension one. I think that would depend, that would be possible, but it would depend a lot on the nature of the infinite dimensions of the universe we're moving inside of, right? Somehow, if Lincoln's real, there's a more universal description of it on the further to the left. This is something that I could even actually work this out because there's a further fact that although there are these many different models of combinatorial, as I said, substantial, cubical, lobular, etc., nonetheless... The dimension of one tends to be reflexive graphs in all these cases. The one-dimensional cubical set is also essentially just a directed graph. So really one should try to work out a more typical case where you have...

1:20:00 Totally abstract sets of dimension zero, rather than the all-important Galois total, which is that it should be dimension zero if you have to break down the field, which is not an algebraic result of the field, and then reflexive graphs as dimension one, and then what are the possible meanings of dimension two, given that constraint, that dimension one is just reflexive graphs, it should be possible to come up with some kind of... More illuminating, more pictorial descriptions, like two co-skeletons. The term, the term skeleton and co-skeleton, may be slightly confusing. You see, you have these, and then on the opposite, you have these punctures going up and one puncture coming down. It takes a space of pi-dimension and strips off everything and makes it look one-dimensional or whatever, two-dimensional at a given level. Now, when I say it makes it look like, what that really means is that you apply the left decoction. You get back into the same category with the central object. It's the endofactor of the general infinite-dimensional space, which is usually called the skeleton. A part is determined in that way, a universal thing on the right from, let's say, one truncation to the same thing on the right, so that regarding the truncation process is more or less obvious and trivial, the means of the postcode is the one that goes up. Which starts with a one-dimensional space and produces an infinite dimension, or starts with a two-dimensional space and produces an infinite dimension. In a sense, when you compose that function with truncations, you take an arbitrary space, truncate, and then go back up, now you get something to which the original space maps, and the codomain is now, in a sense, determined by two-dimensional information, but the domain is arbitrary.

1:22:30 In that sense, the first skeleton of an arbitrary space, but the interesting puncture is that these ones actually start with the low-dimensional space and produce a certain kind of infinite dimensional one, where literally the co-dimensions kind of typically...